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Electronic Transactions on Numerical Analysis.
Volume 6, pp. 35-43, December 1997.
Copyright  1997, Kent State University.
ISSN 1068-9613.
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LOCAL ERROR ESTIMATES AND ADAPTIVE REFINEMENT FOR
FIRST-ORDER SYSTEM LEAST SQUARES (FOSLS)∗
MARKUS BERNDT†, THOMAS A. MANTEUFFEL†, AND STEPHEN F. MCCORMICK†
Abstract. We establish an a-posteriori error estimate, with corresponding bounds, that is valid for any FOSLS
L2 -minimization problem. Such estimates follow almost immediately from the FOSLS formulation, but they are
usually difficult to establish for other methodologies. We present some numerical examples to support our theoretical
results. We also establish a local a-priori lower error bound that is useful for indicating when refinement is necessary
and for determining the initial grid. Finally, we obtain a sharp theoretical error estimate under certain assumptions
on the refinement region and show how this provides the basis for an effective refinement strategy. The local a-priori
lower error bound and the sharp theoretical error estimate both appear to be unique to the least-squares approach.
Key words. adaptive mesh refinement, a-posteriori error estimates, first-order system least-squares.
AMS subject classifications. 65N15, 65N30, 65N50.
1. Introduction. Research has recently intensified in the field of first-order system
least-squares methods (cf. [5], [7], [8], [6], [3], [4] and [11]). Most of the numerical examples presented in these papers are based on uniform grid implementations. However, in
[11], the contribution that an element makes to the total value of the FOSLS functional is
used as a local a-posteriori error estimate in a refinement process. This measure had been
suggested earlier by a number of authors (eg. [12] and [9]), but the reasoning for such an
a-posteriori error estimate has been heuristic. The purpose of the present paper is to put this
methodology on a theoretical footing.
In Section 2, we review the FOSLS methodology to introduce some notation and motivate our later results. In Section 3, we establish the theory for such a local a-posteriori error
estimate using only results from standard FOSLS theory in a general setting. Our results imply that any FOSLS L2 -functional can be used as a local a-posteriori estimate for the error in
the norm that it induces. Such estimates follow almost immediately from the FOSLS formulation, but they can be difficult to establish for other methodologies. We support this theory
and these claims with a numerical example. In Section 4, we show how the FOSLS functional
can also be used to obtain a local a-priori lower error bound. This a-priori bound, which appears to be unique to the least-squares approach, is useful for indicating when refinement is
necessary and determining the initial grid. In Section 5, we develop another unique tool: a
sharp theoretical error estimate. We prove this estimate to be a good indicator of the local
error under certain assumptions on the refinement region, and we show how it provides the
basis for an effective refinement strategy.
2. Notation. In this section, we review the FOSLS methodology in a general setting and
introduce notation used in Section 3.
2.1. The FOSLS Methodology. Here we review the FOSLS methodology. See [5] and
[7] for more detail.
We start with a (typically second-order) partial differential equation, or system of partial
∗ Received
May 29, 1997. Accepted for publication July 31, 1997. Communicated by S. Parter. This work was
sponsored by the National Science Foundation under grant number DMS-9312752, and the Department of Energy
under grant number DE-FG03-94ER25217
† Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder,
Boulder, CO 80303-0526.
M. Berndt (berndt@boulder.colorado.edu), T. A. Manteuffel
(manteuf@boulder.colorado.edu), S. F. McCormick (stevem@boulder.colorado.edu)
35
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Local error estimates and AMR for FOSLS
differential equations:
Lw = f in Ω,
(2.1)
together with appropriate boundary conditions. The FOSLS methodology yields a system of
first-order partial differential equations,
(2.2)
Li u = fi , i = 1 . . . M,
which is equivalent to original problem (2.1). The resulting FOSLS L2 -functional is the
scaled sum of L2 -norms of the residuals of system (2.2):
(2.3)
G(u; f ) =
M
X
ai kLi u − fi k20 , ai > 0.
i=1
(Here we consider the L2 case for definiteness, although the L2 norm k · k0 can be replaced
by any other suitable norm.) The FOSLS minimization problem is
(2.4)
u = arg min G(v; f ),
v∈W
where W is an appropriate Hilbert space, often a product of H 1 spaces. The weak form of
this minimization problem is to find u ∈ W such that
(2.5)
F(u; v) = (f, v)0 , ∀v ∈ W,
where
(2.6)
F(u; v) =
M
X
ai (Li u, Li v)0 .
i=1
The next step in the FOSLS methodology is to establish continuity and coercivity bounds
for bilinear form (2.6) in some suitable norm ||| · |||. In many cases, this norm is a properly
scaled sum of H 1 -norms of components of u. Continuity is
(2.7a)
F(u; v) ≤ c̄ |||u||| |||v|||
and coercivity is
(2.7b)
F(u; u) ≥ c |||u|||2 .
Among other attributes, these bounds imply well-posedness of the FOSLS minimization
problem (2.4).
2.2. The Local FOSLS Functional. The FOSLS functional is a sum of integrals and,
hence, can be evaluated over any subdomain A of Ω. We call
(2.8)
GA (u; f ) =
M
X
ai kLi u − fi k20,A
i=1
the local FOSLS functional. It follows that, for any tessellation T of Ω, the FOSLS functional
can be evaluated as the sum of all local functionals over the elements in the tessellation:
X
(2.9)
G(u; f ) =
Gτ (u; f ).
τ ∈T
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3. A-Posteriori Error Estimates. In the literature, it has been suggested that the value
of the local least-squares functional on an element in a given tessellation can be used as an
a-posteriori error estimate (cf. [12], [9], [11]). From standard FOSLS theory, it is usually
easy to establish bounds for such an a-posteriori error estimate.
For a given uh in some finite-dimensional space W h ⊂ W , define
q
ητ := Gτ (uh ; f )
(3.1)
for any τ ∈ T , where T is any tessellation of the domain Ω (not necessarily associated with
W h ). Now, coercivity bound (2.7b) implies
(3.2)
G(uh ; f ) = G(uh − u; 0) = F(uh − u; uh − u) ≥ c |||uh − u|||2
and, from (2.9) and the definition of ητ , we get
X
X
(3.3)
G(uh ; f ) =
Gτ (uh ; f ) =
ητ2 .
τ ∈T
τ ∈T
Thus, (3.2) and (3.3) imply
(3.4)
|||uh − u|||2 ≤
1 X 2
ητ .
c
τ ∈T
In the literature, an inequality of type (3.4) is called a reliability bound (cf. [13]). If all of the
local error estimates ητ are small, then the error is also small.
For FOSLS formulations, it is typically straightforward to establish bound (2.7a), using
only the triangle, Cauchy-Schwarz, and ε inequalities. Most importantly, we will show in the
next subsection by example that a proof analogous to that for bound (2.7a) generally can be
used to establish a similar bound on the local functional. We get
(3.5)
GA (uh ; f ) = FA (uh − u; uh − u) ≤ c̄|||uh − u|||2A
for any subdomain A of Ω, which implies
(3.6)
|||uh − u|||2τ ≥
1 2
η
c̄ τ
for any τ in T .
For these estimates, we made no assumption about how the approximation uh ∈ W h was
obtained. For standard Galerkin formulations, bounds like (3.4) and (3.6) for an a-posteriori
error estimate usually depend on the fact that uh comes from a discrete finite element solution,
and they can be very tedious to derive (cf. [13] for a number of examples). In contrast, the
choice of (3.1) as an a-posteriori error estimate for a FOSLS formulation is natural, since
bounds (3.4) and (3.6) follow immediately from FOSLS theory. Coercivity bound (2.7b)
yields bound (3.4) as we showed above. A proof analogous to that for continuity bound
(2.7a) yields (3.6), as we now illustrate by two examples.
3.1. Examples. Here we include examples that illustrate how inequality (3.6) can be
established in general. First consider the div-curl functional in 3D:
(3.7)
G(u; f ) = k∇ · u + f k20,Ω + k∇ × uk20,Ω .
To understand our notation, see [7]. From vector calculus and the triangle, Cauchy-Schwarz,
and ε inequalities, we obtain
(3.8a)
k∇ · ek20,A ≤ 3 k∇ek20,A ,
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Local error estimates and AMR for FOSLS
(3.8b)
k∇ × ek20,A ≤ 3 k∇ek20,A .
Here, e ∈ (H 1 (Ω))3 and A is any subdomain of Ω. These two inequalities and the definition
of G imply
(3.9)
ητ2 = Gτ (uh ; f )
= k∇ · (uh − u)k20,τ + k∇ × (uh − u)k20,τ
≤ 6 k∇(uh − u)k20,τ ,
which immediately implies (3.6) with c̄ = 6.
Next, as a more realistic example, consider the Stokes functional discussed in [8]:
(3.10) G(U , u, p; f , g) = kf + ν(∇ · U )t − ∇pk20 + ν 2 kU − ∇ut k20 + ν 2 k∇ × Uk20
+ν 2 k∇ · u − gk20 + ν 2 k∇(trU ) − ∇gk20 .
Similar to the div-curl example, we can establish a bound of type (3.6) using only the triangle,
Cauchy-Schwarz, and ε inequalities:
1 2
η ≤ ν 2 kU h − Uk21,τ + ν 2 kuh − uk21,τ + kph − pk21,τ .
12 τ
These two examples are typical and show that bounds (3.4) and (3.6) follow naturally
from standard FOSLS theory.
(3.11)
3.2. Numerical Results. Here we present a simple heuristic refinement strategy that
incorporates the a-posteriori error estimate (3) to illustrate its practical value. An optimal
strategy determines a refinement region R ⊂ Ω that minimizes the ratio
(3.12)
work to solve the new discrete problem
.
gain in accuracy
Of course, we must somehow approximate both work and gain. For a FOSLS implementation,
the solver of choice is a multilevel algorithm. Hence, a reasonable approximation to work is
a linear function of the number of vertices in the finite element mesh, nold , and the number
of vertices that will be added due to refinement, nnew . The new vertices are added to the
hierarchy of levels as the finest level. Hence, the work induced by these points (relaxation and
grid transfers) is proportional to nnew . The work induced by the old vertices is proportional
to nold , since a multigrid cycle is used to solve the coarse level problem. Thus, the following
approximation for work is used:
(3.13)
work to solve the new discrete problem ≈ anold + bnnew ,
with suitable constants a and b. The gain in accuracy can be measured by calculating the ratio
(3.14)
gain in accuracy =
G(uhold ; f )
.
G(uhnew ; f )
We cannot calculate G(uhnew ; f ) before refining the mesh, which we assume is accomplished
by halving the mesh size: mesh size h is reduced to h2 . For a FOSLS discretization of order
h, we typically get bounds like |||e||| ≤ h |||∇u|||, so cutting the mesh size h in half tends to
reduce the functional by a factor of 4. Thus, we use the approximation
1
GR (uhold ; f ) + GRc (uhold ; f ).
4
Our refinement strategy is as follows.
(3.15)
G(uhnew ; f ) ≈
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1. Find the maximum of the a-posteriori error estimates over all τ ∈ T : ητ,max .
i 2
2
2. Partition T into contour sets Ci = {τ |ητ2 ∈ ( i−1
N ητ,max , N ητ,max ]}, i =
1, 2, . . . , N .
3. Calculate wi = work/gain, for i = 1, 2, . . . , N .
4. Find i ∈ {1, 2, . . . , N } for which wi is minimal.
5. Refine contour sets Ci . . . CN .
We applied this refinement strategy to the FOSLS minimization problem involving functional (3.7) on the unit square with boundary conditions τ · u = 0 (which corresponds with
u = ∇p to a Poisson equation with homogeneous Dirichlet boundary conditions). In this
example, we chose f = χB0.1 ( 34 , 34 ) , the characteristic function of a ball of radius 0.1 centered at the point ( 34 , 34 ). We used continuous piecewise linear elements that conformed to the
boundary conditions. Table 3.1 shows the value of the global FOSLS functional after solvG(u; f )
G(u; f )
(loc. ref.) (glob. ref.)
0
62
8.05(-3)
8.05(-3)
1
94
7.38(-3)
7.51(-3)
2
214
3.99(-3)
4.68(-3)
3
468
2.14(-3)
3.59(-3)
4
866
1.36(-3)
2.75(-3)
5
2580
6.56(-4)
1.64(-3)
6
4705
3.80(-4)
1.15(-3)
TABLE 3.1
Value of FOSLS functional: local versus global refinement
iteration
# unknowns
ing the minimization problem on the locally refined finite element mesh. As a reference, we
also show the value of the global FOSLS functional after solving on a uniform finite element
mesh with approximately the same number of vertices as the corresponding refined mesh.
Our results show that the a-posteriori error estimate (3.1) is a good indicator for the necessity
of refinement.
4. A Local A-Priori Lower Error Bound. Here we show how the local FOSLS functional can be used to obtain a local a-priori error estimate.
For a given element τ , define the estimate
r
(4.1)
η̄τ :=
min Gτ (vh ; f ).
vh ∈W h
Then, for any tessellation T that contains τ , we have
(4.2)
η̄τ2 ≤ Gτ (vh ; f ), ∀vh ∈ WTh .
Using (3.6), this implies that
(4.3)
η̄τ2 ≤ ητ2 ≤ c̄ |||uh − u|||2τ
for any tessellation T with its associated discrete space WTh .
A-priori error estimate η̄τ can be used to obtain a good initial tessellation of the domain
Ω. Using bound (4.2), we can calculate a global a-priori lower bound for the value of the
FOSLS functional and, hence, for the error on a given tessellation T :
X
(4.4)
η̄τ2 ≤ G(uh ; f ).
τ ∈T
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Local error estimates and AMR for FOSLS
If the goal is to start the adaptive algorithm with an initial tessellation T that is fine enough
to resolve the right-hand side (see for example [10], section 5.1), one could use a-priori lower
bound (4.2) as an indicator of where to refine the initial tessellation. The following algorithm
could be used to obtain such an initial tessellation:
1. Construct a tessellation T0 that resolves the geometry of Ω well.
2. Given a global lower bound κ > 0 on the FOSLS functional
P that is acceptable, refine
T0 adaptively until the refined tessellation T1 satisfies τ ∈T1 η̄τ2 ≤ κ.
Note that such an algorithm can also be useful for subsequent ’initial’ tessellations
involved in an FMG-like process of successively decreasing values of κ. Such a ’κcontinuation’ process could be useful in the computation of an initial solution approximation
with some final value of κ.
The evaluation of η̄τ can be done by solving a local least-squares minimization problem
which involves inverting the individual element stiffness matrices. Calculating η̄τ for every τ
is, thus, comparable in computational cost to a point relaxation sweep (where all variables at a
point are relaxed simultaneously). On a structured finite element mesh, there are only a small
number of different element stiffness matrices (in the case of piecewise-constant coefficients),
so the calculation of η̄τ can be done very efficiently. Also, the information needed to evaluate
η̄τ is entirely local to an element τ , so the a-priori error estimate can be calculated in parallel.
5. A Sharp Theoretical Estimate and Convergence of an Adaptive Algorithm. In
this section, we present a theoretical result regarding error reduction in an adaptive algorithm
for FOSLS. We first introduce some notation.
Suppose that uh is the best approximation to the solution u of minimization problem
(2.4) on the current level W h . Let R ⊂ W be a subregion in which further refinement is
considered. Define the error e = uh − u, which we decompose into ‘local’ and ‘harmonic’
parts as follows. First, let the set of local functions (with support in R) be defined by
(5.1)
WR := {v ∈ W : v = 0 on Rc ≡ Ω − R}.
Then, define
(5.2)
l := arg min G(e + v; 0).
v∈WR
Now let
h = e − l,
(5.3)
and note that h = e on Rc . We say that h is locally F-harmonic on R since
(5.4)
F(h; v) = 0, ∀v ∈ WR .
Note in particular that e = h + l is an F-orthogonal decomposition of e in the sense that
(5.5)
F(h; l) = 0.
Note also that we can equivalently define h as follows:
(5.6)
h := arg
min
v=e on Rc
GR (v; 0).
The harmonic h is the component of the error that cannot be eliminated even with infinite
refinement of R. On the other hand, the local error l is the component of the error that can
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41
be fully resolved by infinitely refining in R and leaving the approximation in Rc fixed. Our
principle theorem shows that the FOSLS functional can be used to identify large local error
provided the local region is not too ’thin’.
T HEOREM 5.1. Given region R ⊂ Ω, approximation uh ∈ W h , and error decomposition (5.3), define ε by
GR (e; 0) = (1 − ε) G(e; 0).
(5.7)
Assume that there exists γ < 1 − ε such that
GR (h; 0) ≤ γ G(h; 0).
(5.8)
Then
G(h; 0) ≤ δ G(e; 0),
(5.9)
ε
where δ = 1−γ
< 1.
Proof. Inequalities (5.7) and (5.8) imply
(1 − γ) G(h; 0) = G(h; 0) − γ G(h; 0)
≤ G(h; 0) − GR (h; 0)
= GRc (e; 0)
= G(e; 0) − GR (e; 0)
≤ G(e; 0) − (1 − ε) G(e; 0)
= ε G(e; 0).
Hence,
G(h; 0) ≤
ε
G(e; 0),
1−γ
which completes the proof.
R EMARK 1. Under the assumptions of Theorem 5.1, it is always possible to choose
γ ≤ 1 − ε, with strict inequality possible when the local error l is nonzero.
Proof. We have
GR (h; 0) ≤ GR (e; 0) =
(5.10)
1−ε
GRc (e; 0).
ε
Adding to this the tautology
1−ε
1−ε
GR (h; 0) =
GR (h; 0),
ε
ε
(5.11)
we get
(5.12)
(1 +
1−ε
1−ε
)GR (h; 0) ≤
(GRc (e; 0) + GR (h; 0)),
ε
ε
or
(5.13)
GR (h; 0) ≤ (1 − ε)G(h; 0).
Hence, we can always choose γ in (5.8) to be less or equal to 1 − ε. Note that the inequality
in (5.10) is strict when l 6= 0, in which case we may choose γ < 1 − ε.
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Local error estimates and AMR for FOSLS
We have not used any specific information about G, so Theorem 5.1 holds for any functional. We have also not yet used any information about how uh was obtained, so this estimate
really holds for any approximation in W .
Equality (5.7) can be interpreted in the following way. We want to choose a refinement
region R ⊂ Ω that contributes a significant portion to the total value of the functional. The
idea is to identify a subregion R of Ω so that ε defined by (5.7) is fairly small. Inequality
(5.8) is a statement about the shape of the refinement region R. If R is too ‘thin’, then the
constant γ will be close to 1, in which case a lot of energy (i.e., value of G) can be hidden
(inaccessible to refinement) in the F-harmonic part of the error. Hence, we want to choose a
refinement region that is not too small (nor too large or else global refinement would be more
efficient). Inequality (5.9) means that exact solution of the minimization problem in R would
reduce the total value of the functional by at least a factor of δ. The theorem thus asserts that,
under certain restrictions on the refined region, the FOSLS functional can be a sharp predictor
of the error that refinement would eliminate.
In practice, infinite refinement is generally impossible. To illustrate how Theorem 5.1 can
be applied when only one refinement level is used in R, assume that refinement by halving h
reduces the local error l in R by a factor of 14 :
(5.14)
1
G(uh/2 ; f ) ≤ GRc (e; 0) + GR (h; 0) + GR (l; 0).
4
Here uh/2 is equal to the initial approximation, uh , on R̄c , but it is obtained by solving the
local FOSLS minimization problem on R. Consider the relations
(5.15)
GR (l; 0) = GR (e; 0) − GR (h; 0),
which follows from (5.5), and
(5.16)
GR (h; 0) ≤
γ
GRc (e; 0),
1−γ
which follows from (5.8) and the fact that h = e on Rc and G(h; 0) = GR (h; 0) + GRc (h; e).
Then (5.14), (5.15), (5.16), and (5.7) imply
1
(GR (e; 0) − GR (h; 0))
4
3
1
= GRc (e; 0) + GR (h; 0) + GR (e; 0)
4
4
3 γ
1
≤ 1+
GRc (e; 0) + GR (e; 0)
41−γ
4
3 γ
1
= 1+
ε G(e; 0) + (1 − ε)G(e; 0)
41−γ
4
1 3 ε
=
+
G(e; 0).
4 41−γ
G(uh/2 ; f ) ≤ GRc (e; 0) + GR (h; 0) +
ε
Thus, we obtain a bound similar to (5.9), but now with δ = 14 + 34 1−γ
. This implies converε
gence for the case of one additional level of refinement when 1−γ < 1 (i.e., l 6= 0).
6. Conclusions. We want to stress that the bounds on the a-posteriori error estimate ητ
follow immediately from standard FOSLS theory. This is a substantial advantage over what
has been obtained for other methods (see, for example, [1], [2], or, for a review of a-posteriori
error estimators, [13]). We have also showed how FOSLS naturally yields a local a-posteriori
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error estimate. The numerical results presented in Section 3.2 support the practicality of the
proposed a-posteriori error estimate for adaptively refining a finite element mesh.
The local a-priori lower error bound η̄τ can be an important tool in the context of adaptive
refinement, either for generating an initial mesh or for use in the process of adaptively refining
a computational mesh. An important strength of this measure is that an unacceptably large
value η̄τ is a certain signal that τ must be refined.
Theorem 5.1 shows that the FOSLS functional provides a sharp measure of the local
error, provided estimate (5.8) holds. This restriction is related to the geometry of R and the
nature of the specific FOSLS functional. To confirm (5.8), we need to be sure that local Fharmonics do not exhibit inordinately large local energies (otherwise, the refinement process
could be misled by large local F-harmonic errors, which cannot be eliminated within the
local region itself). What is needed is an articulation of the assumptions on G and specific
restrictions on R that could be used to guide the refinement process. This should lead to a
proof of convergence of the refinement strategy that is potentially sharper and more general
than other theories (see [10] for important recent work on Poisson’s equation).
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